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Quantum cohomology and Fukaya summands from monotone Lagrangian tori

Jack Smith

TL;DR

This work establishes a precise link between the quantum cohomology of a compact symplectic manifold X and the superpotential W_L of monotone Lagrangian tori L. By analysing the closed–open map and localising at isolated critical points, the authors prove that QH^*(X) decomposes as a product with a factor isomorphic to Jac_isol W_L, and that the corresponding Fukaya summands are generated by toroidal L^flat objects, often described by matrix-factorisation categories. The paper provides an automatic split-generation mechanism for toroidal subcategories, relates Hochschild and quantum cohomology via wpCY structures, and gives a geometric realisation of these decompositions through disc counts, thereby tying mirror symmetry predictions to explicit algebraic decompositions. Through worked examples—the cubic surface, the quadric threefold, and the 4-point CP^2 blowup—the results yield concrete constraints on possible superpotentials W_L and illuminate how the global QH^*(X) structure constrains monotone tori. Overall, the work advances understanding of how monotone Lagrangians control the algebraic structure of quantum cohomology and its mirror-theoretic interpretations.

Abstract

Let $L$ be a monotone Lagrangian torus inside a compact symplectic manifold $X$, with superpotential $W_L$. We show that a geometrically-defined closed-open map induces a decomposition of the quantum cohomology $\operatorname{QH}^*(X)$ into a product, where one factor is the localisation of the Jacobian ring $\operatorname{Jac} W_L$ at the set of isolated critical points of $W_L$. The proof involves describing the summands of the Fukaya category corresponding to this factor -- verifying the expectations of mirror symmetry -- and establishing an automatic generation criterion in the style of Ganatra and Sanda, which may be of independent interest. We apply our results to understanding the structure of quantum cohomology and to constraining the possible superpotentials of monotone tori

Quantum cohomology and Fukaya summands from monotone Lagrangian tori

TL;DR

This work establishes a precise link between the quantum cohomology of a compact symplectic manifold X and the superpotential W_L of monotone Lagrangian tori L. By analysing the closed–open map and localising at isolated critical points, the authors prove that QH^*(X) decomposes as a product with a factor isomorphic to Jac_isol W_L, and that the corresponding Fukaya summands are generated by toroidal L^flat objects, often described by matrix-factorisation categories. The paper provides an automatic split-generation mechanism for toroidal subcategories, relates Hochschild and quantum cohomology via wpCY structures, and gives a geometric realisation of these decompositions through disc counts, thereby tying mirror symmetry predictions to explicit algebraic decompositions. Through worked examples—the cubic surface, the quadric threefold, and the 4-point CP^2 blowup—the results yield concrete constraints on possible superpotentials W_L and illuminate how the global QH^*(X) structure constrains monotone tori. Overall, the work advances understanding of how monotone Lagrangians control the algebraic structure of quantum cohomology and its mirror-theoretic interpretations.

Abstract

Let be a monotone Lagrangian torus inside a compact symplectic manifold , with superpotential . We show that a geometrically-defined closed-open map induces a decomposition of the quantum cohomology into a product, where one factor is the localisation of the Jacobian ring at the set of isolated critical points of . The proof involves describing the summands of the Fukaya category corresponding to this factor -- verifying the expectations of mirror symmetry -- and establishing an automatic generation criterion in the style of Ganatra and Sanda, which may be of independent interest. We apply our results to understanding the structure of quantum cohomology and to constraining the possible superpotentials of monotone tori
Paper Structure (23 sections, 31 theorems, 89 equations, 1 figure)

This paper contains 23 sections, 31 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Mirror symmetry picture
  4. Constraints on quantum cohomology from monotone tori
  5. Constraints on monotone tori from quantum cohomology
  6. Isolated critical points
  7. Structure of the paper
  8. Acknowledgements
  9. Generation from smoothness
  10. Split-generation
  11. The Shklyarov pairing
  12. The closed--open and open--closed string maps
  13. The weak proper Calabi--Yau structure
  14. Decompositions and the generation
  15. Proof of the main results
  16. ...and 8 more sections

Key Result

Theorem A

As an $S_\mathrm{isol}$-algebra, $\operatorname{HF}^*_\mathrm{isol}(\mathbf{L}, \mathbf{L})$ is generated by the unit and is equal to $\operatorname{Jac}_\mathrm{isol} W_L$. The $\mathbb{Z}/2$-graded unital $\mathbf{k}$-algebra homomorphism is surjective, and gives rise to an orthogonal decomposition (of $\mathbb{Z}/2$-graded unital $\mathbf{k}$-algebras) with respect to the Poincaré pairing. He

Figures (1)

  • Figure 1: Mutations and changes of variable.

Theorems & Definitions (70)

  • Example 1.1
  • Theorem A: \ref{['corThmA']}
  • Corollary B
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Remark 1.5
  • Theorem C: \ref{['propCategoryDescriptions']}
  • Definition 1.6
  • Theorem D: \ref{['propSegal']}
  • ...and 60 more